Scaling H-like Functions
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Scaling Size with Z
p=(2Z)/(na)*r
- The triple equals sign means "is defined as".
- Z is the nuclear charge or atomic number
- n is the principle quantum number
- a is Bohr's radius of the H atom
- r is the radius, and comes from the wavefunction
- Therefore, p is a scaled r
From this equation, we can see that increasing Z will make the radius smaller for the same p. This is shown visually in Slide 12, which shows the radii of the 1s shells for 1-electron H, C, and K.
[edit]Scaling Electron Desity with Z
Electron density (psi^2) also depends on Z. The wavefunctions (psi) contain (Za)^(3/2); when this is squared to get the electron density, we get Z^3. Increasing Z will increase the density by Z^3. This is the relationship between radius and volume of a sphere, which makes sense since we are calculating the density over all space (volume). X-rays are able to find heavier atoms more easily because they have a higher electron density.
We also have to normalize the psi^2 function to 1 (which gives us all the nasty constants), otherwise the sum of the probability densities will not add to 1, and 1 is the highest value a probability can take.
[edit]Scaling Kinetic Energy with Z
Recall that KE= -C/m*(curvature of psi)/psi
Curvature of psi is its second derivative.
F(Z*r) is a psi function of a scaled r. Differentiating the function twice (and remembering the chain rule) gives curvature=Z^2*F(Z*r). So increasing Z increases the curvature (and thus the kinetic energy) by Z^2.
[edit]Scaling Potential Energy with Z
Coulomb's Law gives V=Ze/r.
- V is the potential energy
- e is a constant
By this equation, increasing Z will also increase V by a factor of Z, if the shape of the orbital is not changed. However, because we now have a higher atomic number, the radius will shrink by Z (as shown above). The potential energy is being affected by Z in two different ways (through Coulomb's Law and higher effective nuclear charge), so it increases by Z^2.
[edit]Scaling Total Energy with Z
Total energy= -RZ^2/n^2
- R is a constant, approx. 300 kcal/mol
n^2 in the denominator makes this an inverse square law. As n increases, the distance between consecutive energies gets smaller. This is shown by the lines on the upper-left0hand side of the slide.
From this equation, we again see that increasing Z will increase the energy by Z^2.
We also notice that the quantum numbers l and m do not appear in the formula for total energy. Total energy only depends on the principle energy level.
[edit]Summary
- Size is proportional to 1/Z or n/Z
- Electron density is proportional to Z^3
- Energy is proportional to Z^2 or Z^2/n^2
